Armed with the self-improvement lemma, we set up the following
variant of the induction loop sued in [1]}. We say that an affine subspace \(S\)  is “\(k\) -aligned” if \(S\)  has dimension \(k\)  and for all \(u,v \in S\) , the projection of \(u,v\)  to the first \(m-k\)  coordinates are the same. For \(k\)  running from 1 to \(m\) ,
for all \(k\) -aligned subspaces \(S\) , we construct a “subspace measurement” \(\lbrace G^{S}_{g}\rbrace \)  that returns a codeword \(g \in \mathcal {C}^{\otimes k}\) , as follows.
